北京雁栖湖应用数学研究院

Based on joint work with Davide Macera and Joe Thomas. The first non-zero eigenvalue, or spectral gap, of the Laplacian on a closed hyperbolic surface encodes important geometric and dynamical information about the surface. We study the size of the spectral gap for random large genus hyperbolic surfaces sampled according to the Weil–Petersson probability measure. We show that there is a $c>0$ such that a random surface of genus $g$ has spectral gap at least $1/4-O(g^{-c})$ with high probability. Our approach adapts the polynomial method for the strong convergence of random matrices, introduced by Chen, Garza-Vargas, Tropp and van Handel, and its generalization to the strong convergence of surface groups by Magee, Puder and van Handel, to the Laplacian on Weil–Petersson random hyperbolic surfaces.

In this talk, I will first recall some relevant notations and properties of the Brooks–Makover model of random Belyi surfaces. l will then present our recent result on the spectral gap in this model. We show that, as $n$ tends to infinity, a random hyperbolic surface in the Brooks–Makover model has a spectral gap greater than $\frac{1}{4}-\frac{1}{n^{1/221}}$. This is joint work with Yunhui Wu.

We investigate uniform spectral gaps for Weil–Petersson random hyperbolic surfaces with not many cusps. We show that if $n=O(g^\alpha)$ where $\alpha\in \left[0,\frac{1}{2}\right)$, then for any $\epsilon>0$, a random cusped hyperbolic surface in $\mathcal{M}_{g,n}$ has no eigenvalues in $\left(0,\frac{1}{4}-\left(\frac{1}{6(1-\alpha)}\right)^2-\epsilon\right)$. If $\alpha$ is close to $\frac{1}{2}$, this gives a new uniform lower bound $\frac{5}{36}-\epsilon$ for the spectral gaps of Weil–Petersson random hyperbolic surfaces. The major contribution of this work is to reveal a critical phenomenon of "second order cancellation". This talk is based on a joint work with Yunhui Wu and Yuhao Xue.

In recent years, the strong convergence properties of group representations of fundamental groups of graphs and manifolds have been successfully exploited to obtain near optimal spectral gap results for their Laplacians. Proving the existence of sequences of unitary representations that strongly converge to the regular representation for different groups is a difficult task with few concrete results. In this talk, I will discuss joint work with Michael Magee (Durham) where we prove the existence of strongly converging unitary representations for right-angled Artin groups. These groups are important in geometric group theory and closely related to the fundamental groups of closed hyperbolic 3-manifolds. I will focus on this connection and explain the spectral implications.

In this talk, we discuss extremal problems for Laplace–Beltrami eigenvalues on unit-area tori. We first review Berger’s isoperimetric problem for the first eigenvalue and present a new proof using the conformal area method and Bryant’s work on minimal immersions. We then study the second eigenvalue. Building on recent work of Karpukhin–Stern and Eddaoudi–Girouard, we establish an explicit upper bound for $\lambda_2$ in any fixed conformal class of a torus. Our result improves previous estimates and provides further evidence for the Kao–Lai–Osting conjecture.

We show that for any fixed $k \geq 1$, as the genus goes to infinity, the maximum of $\lambda_{k}-\lambda_{k-1}$ over any thick part of the moduli space of closed Riemann surfaces approaches the limit $\frac{1}{4}$. This talk is based on a joint work with Yunhui Wu.